Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
A monoidal model category is a model category which is also a closed monoidal category in a compatible way. In particular, its homotopy category inherits a closed monoidal structure.
A (symmetric) monoidal model category is model category equipped with the structure of a closed symmetric monoidal category such that the following two compatibility conditions are satisfied
(pushout-product axiom) For every pair of cofibrations and , their pushout-product, hence the induced morphism out of the cofibered coproduct over ways of forming the tensor product of these objects
is itself a cofibration, which, furthermore, is acyclic if or is.
(Equivalently this says that the tensor product is a left Quillen bifunctor.)
(unit axiom) For every cofibrant object and every cofibrant resolution of the tensor unit , the resulting morphism
is a weak equivalence.
(Schwede-Shipley 00, def. 3.3.).
Monoidal homotopy category
Let be a monoidal model category, and consider it as a derivable category in the sense of (Shulman 11, section 8) with the subcategory of cofibrant objects and . Then is left derivable, i.e. it preserves the -subcategories and weak equivalences. Since deriving is pseudofunctorial (and product-preserving) on the 2-category of derivable categories and left derivable functors, it follows immediately that is monoidal; this is (Shulman 11, example 8.13).
Now let denote the category with its trivial derivable structure: only isomorphisms are weak equivalences, and all objects are both and . Then of course , and is also a pseudomonoid in derivable categories. The identity functor is not left derivable, since it does not preserve -objects; but it is right derivable, since we took all objects in to be -objects (ignoring the fibrant objects in the model structure on ). Of course is strong monoidal, and this monoidality constraint can be expressed as a square in the double category of (Shulman 11) whose vertical arrows are left derivable functors and whose horizontal arrows are right derivable functors; moreover the axioms on a monoidal functor may be expressed using products and double-categorical pasting in this double category. Therefore, it is all preserved by the double pseudofunctor ; but . The only thing that is not visible to the double category is the invertibility of the monoidal constraint, and hence this is not preserved by the double pseudofunctor; thus is only lax monoidal.
Let be a monoidal model category with cofibrant tensor unit. Then the left derived functor of the tensor product exists
and makes the homotopy category into a monoidal category .
Moreover, the localization functor
on the category of cofibrant objects is a strong monoidal functor with structure morphism the inverse of the above natural isomorpmism
For the left derived functor (def.) of the tensor product
to exist, it is sufficient that its restriction to the subcategory
of cofibrant objects preserves acyclic cofibrations (Ken Brown's lemma, here).
Every morphism in the product category may be written as a composite of a pairing with an identity morphisms
Now since the pushout product (with respect to tensor product) with the initial morphism is equivalently the tensor product
the pushout-product axiom (def. 1) implies that on the subcategory of cofibrant objects the functor preserves acyclic cofibrations. (This is why one speaks of a Quillen bifunctor, see also Hovey 99, prop. 4.3.1).
By the same decomposition and using the universal property of the localization of a category (def.) one finds that for and any two categories with weak equivalences (def.) then the localization of their product category is the product category of their localizations:
With this, the universal property as a localization (def.) of the homotopy category of a model category (thm.) induces associators: Let
be the natural isomorphism exhibiting the derived functors of the two possible tensor products of three objects, as shown at the top. By pasting the second with the associator natural isomorphism of we obtain another such factorization for the first, as shown on the left below,
and hence by the universal property of the factorization through the derived functor, there exists a unique natural isomorphism such as to make this composite of natural isomorphisms equal to the one shown on the right. Hence the pentagon identity satisfied by implies a pentagon identity for , and so is an associator for .
The above equation on pasting composites of natural isomorphism is equivalently just the coherence law for a monoidal functor:
Classical homotopy theory
Homological algebra and stable homotopy theory
With respect to a symmetric monoidal smash product of spectra:
(Schwede-Shipley 00, MMSS 00, theorem 12.1 (iii) with prop. 12.3)
The standard example of a monoidal model category whose unit is not cofibrant is the category of EKMM S-modules.
Categorical model structures
Model structure on -objects
Let be a category equipped with the structure of
Under these conditions there is for each finite group the structure of a monoidal model category on the category of objects in equipped with a -action, for which the forgetful functor
preserves and reflects fibrations and weak equivalences.
See for instance (BergerMoerdijk 2.5).
Model structure on monoids
See model structure on monoids in a monoidal model category.
A textbook reference is
Some relevant homotopy category background is in
The monoidal structures for a symmetric monoidal smash product of spectra are due to
Stefan Schwede, Brooke Shipley, Algebras and modules in monoidal model categories Proc. London Math. Soc. (2000) 80(2): 491-511 (pdf)
Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, part III of Model categories of diagram spectra, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 (pdf, publisher)
Further variation of the axiomatics is discussed in
The monoidal model structure on is discussed for instance in
Relation to symmetric monoidal (infinity,1)-categories is discussed in